Astronomy & Celestial Mechanics
the sky, computed
Concept

Tidal locking — where it appears

The end state in which a body's rotation has been driven into commensurability with its orbit by tidal torque. A circular orbit gives the one-to-one lock the Moon shows, an eccentric one can give a higher ratio, and an irregular body at moderate eccentricity may lock into nothing at all.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

The tidal field is a difference. The pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one.

The tide is a difference, which is why there are two of them

The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.

gravitation · Tides The tide across a moon, against the moon's own gravity. The tidal acceleration across a satellite and the satellite's own surface gravity, both in units of that surface gravity, against distance from the primary in planet radii. The tide falls as the inverse cube and the self-gravity does not fall at all, so they cross once — at 2.23 radii for the density ratio drawn. Inside the crossing the tide wins and a body held together only by its own weight comes apart.

The distance at which a moon stops holding together

The tide across a body falls as the inverse cube; the body's own gravity does not fall at all. There is therefore exactly one crossing, and Saturn's rings end within a few per cent of it.

gravitation · Tides Integrating the measured recession back: the Moon reaches the Earth 1.54 Gyr ago. The Earth–Moon separation and the length of the Earth's day, integrated backwards from the measured present recession rate of 3.83 cm per year. Constant-Q tidal friction makes a^(13/2) linear in time, so the history is a single line in a variable nobody plots, and it is calibrated to the laser-ranging measurement rather than to a modelled k₂/Q — the k₂/Q it implies is 0.0257, or Q = 11.6 for the Earth's k₂ of 0.299, which is a startlingly dissipative Earth. Run back at that rate the separation reaches zero 1.54 Gyr ago and crosses the Roche limit at 2.88 Earth radii only 4 years before it, so the drawing is cut off there rather than extrapolated. The Moon is 4.5 Gyr old, so this is a refutation and not a date: the present rate cannot have been the rate, and a mean Q of 34 — drawn dashed, reaching 4.51 Gyr — is the sort of value the age requires. Tidal rhythmites at 620 Myr put the day at 21.9 h and the Moon at 96.5 per cent of its present distance, and this history reads 20.1 h and 92.4 per cent — too fast and too close, which is the same failure the zero crossing is. Day length follows from total angular momentum, 23.93 h today, 9.84 h at half the present lunar distance and 4.97 h at the Roche limit, and depends on the separation alone: it is the same curve whatever Q is. The rate of lengthening the recession requires is 2.10 ms per century, against a tidal total of about 2.3 including the Sun's tide, which slows the Earth without moving the Moon, and an observed 1.75 from ancient eclipses and occultations — the shortfall being the Earth's moment of inertia falling as the mantle rebounds from the last glaciation.

A day five hours long

The tidal bulge leads, so the Earth's spin is being paid into the Moon's orbit. Run the measured payment backwards and two curves come out of one integration — a timeline that is refuted by the Moon's own age, and a day length that is refuted by nothing.

gravitation · Tides The sub-Earth point over 400 days. Where on the Moon the Earth stands overhead, in selenographic longitude and latitude, sampled over 400 days. The excursion in longitude is the equation of centre — the Moon's rotation is uniform and its orbital motion is not, so the face runs alternately ahead and behind by up to 6.3° — and the excursion in latitude is the tilt of its equator, up to 6.7°. The two run on months of different length, so the track never repeats.

The face that is not quite fixed

The Moon keeps one face turned toward the Earth, and the sentence is exactly true only of a fictitious Moon on a circular orbit. The real one rocks by a few degrees each month, in two directions and for two unrelated reasons, and the rocking has shown 59 per cent of its surface to people who never left the ground.

sky · Libration The habitable zone, and the radius inside which a day never ends. The conservative habitable zone — the runaway-greenhouse and maximum-greenhouse limits of Kopparapu's parameterisation — as a band in stellar mass against orbital distance, both logarithmic. For the Sun it runs from 0.99 to 1.71 AU, which is the published result and the check on the arithmetic here. The band moves inward far faster than the mass falls, because luminosity goes as roughly the fourth power of mass: a 0.2 M☉ star's zone is at 0.082–0.158 AU. The dashed line is the distance inside which a planet is tidally locked within 4.5 billion years, computed from τ = ω₀αmQa⁶/(3GM⋆²k₂R³) with an initial ten-hour spin, Q = 100 and k₂ = 0.3. It crosses the inner edge of the zone at 0.67 M☉ — so around every star below that, which is the great majority of stars, a planet in the habitable zone has one hemisphere in permanent daylight. The Earth is outside its own locking radius of 0.53 AU, and TRAPPIST-1e at 0.029 AU is inside its star's by a factor of 8.

One number sets the zone

The habitable zone is a band of stellar flux, so it scales as the square root of luminosity and moves inward far faster than mass falls. For most stars it lies inside the radius at which a planet is tidally locked.

exoplanets · Habitable zone One whole orbit. The system's total brightness through one orbit: the transit at phase 0, the slow rise and fall of the planet's illuminated hemisphere between, and the secondary eclipse at phase 0.5 where the planet's own light is removed. The transit is 1.05%; the secondary eclipse is 1800 ppm, about 6 times shallower.

The planet is seen when it disappears

Half an orbit after the transit the planet passes behind its star, and the light that vanishes is the planet's own. Subtracting two brightnesses taken hours apart isolates a body nothing has ever resolved.

exoplanets · Exoplanet atmospheres The habitable zone of a 1 M☉ star, sweeping outwards. The inner and outer edges of the liquid-water zone against time, for a 1 solar-mass star whose main sequence lasts 10.0 Gyr. The star brightens as it burns hydrogen — a heavier core needs a hotter centre to hold the star up — so both edges move outward by a factor of 1.63 across the whole main sequence, and the band drawn here sweeps past any fixed orbit rather than containing it. Two quite different zones can be read off. The instantaneous zone at the age of the present-day Sun is 0.99 to 1.71 AU, which is the band a survey means by "in the habitable zone". The continuously habitable zone over the 10.0 Gyr drawn is the overlap of every instant in it — outside the inner edge at the end and inside the outer edge at the beginning — which is 1.37 to 1.44 AU, 10 per cent of the instantaneous width. The horizontal line is an orbit at 1 AU. It leaves the zone at 4.72 Gyr, when the inner edge overtakes it. None of these edges is a measurement: both come from one-dimensional climate models, and the inner one in particular is where a runaway greenhouse begins in a model whose clouds are prescribed.

The band moves and the orbit does not

A star brightens as it burns, so the distance at which water can be liquid sweeps outwards by a factor of one and a half across a main sequence. The band a survey quotes is an instant; the band a planet needs is the overlap of every instant, and for the Sun it is a tenth as wide and does not contain the Earth.

exoplanets · Habitable zone Where the gradient of gravity holds a spacecraft still. The plane of the two inertia ratios that decide whether a spacecraft pointing at the Earth is held there by the gravity gradient: k₁ — the pitch moment of inertia less the yaw moment, divided by the roll moment — across, and k₃ — the pitch moment less the roll moment, divided by the yaw moment — up, with roll along the velocity, pitch normal to the orbit and yaw towards the Earth. Shaded points satisfy all three conditions of the linear theory — pitch is stable when k₁ > k₃, and roll and yaw together when k₁k₃ > 0 and 1 + 3k₁ + k₁k₃ > 4√(k₁k₃). The large region at upper right, 12.4 per cent of the square, is the one in which the pitch moment is the largest and the yaw moment the smallest, the arrangement of a long boom hanging towards the Earth. The small region just left of the vertical axis and below the horizontal one, 2.0 per cent, is a second, narrow island of stability with the moments in a different order, found by DeBra and Delp in 1961. There the orientation is a maximum of the potential in roll and yaw rather than a minimum, held only by the gyroscopic coupling of the two, and a damper — the very thing the long-boom region needs — destroys it: with damping of 0.05 of the orbital rate, a swing of a hundredth of a radian at (−0.10, −0.21) grows to a full radian within 11 orbits, while the same swing on the long boom shrinks 435-fold in 40. A boom along the vertical, pitch moment largest sits at (0.97, 0.40) and is stable; the same boom with roll and pitch moments swapped sits at (0.93, −0.40) and is unstable: the same boom, with two nearly equal moments exchanged, crosses from one side of an axis to the other.

A boom held upright by a difference in gravity

A long spacecraft in orbit is pulled into line with the vertical for nothing — its near end feels slightly more gravity than its far end, and the difference is a torque. The torque restores and never dissipates, so the vehicle swings like a pendulum whose clock is the orbit. Whether it is held at all comes down to three inequalities between its moments of inertia, and one region that satisfies all three is destroyed by the damper every such spacecraft needs.

spaceflight · Attitude control A solar day that runs off to infinity as the spin approaches the orbit. The length of the solar day — sunrise to sunrise — in units of the world's own year, against its sidereal rotation period in the same units, with retrograde spin to the left. A world that spins much faster than it orbits has a solar day barely longer than its rotation, which is the Earth's case: its rotation is a few thousandths of its year, and it sits at the foot of the curve just right of zero, too close to the axis to mark. As the spin slows towards the orbital rate the solar day diverges, because the world turns to follow the Sun: at exactly one rotation per orbit the Sun stands still in its sky and there is no day at all. Mercury, turning three times in two of its years, has a solar day of 2.000 years — 176 Earth days. Venus, turning slowly backwards, has one of 116.8 days, 0.52 of its years, with the Sun rising in the west. The definitions of a day and of a year that the Earth keeps far apart come within a factor of two of each other, or exchange places.

A day longer than the year

On the Earth a day and a year are far apart, so a calendar has only to count one in the other. On a world whose spin is not much faster than its orbit, the solar day runs off towards infinity: Mercury's day lasts two of its years, Venus's sun rises in the west twice a year, and a locked moon has a day set by its planet. Every calendar problem met on the Earth returns, and some of them return inverted.

sky · Calendars Four rocky planets against the bare-rock limit: TRAPPIST-1b at 0.99, TRAPPIST-1c at 0.87, LHS 3844b at 1.01, GJ 367b at 0.99. Published day-side brightness temperatures of four small planets orbiting M dwarfs, each divided by the temperature its day side would have as dark bare rock re-radiating where it absorbs (dot, with its uncertainty), with the value for heat spread evenly over the whole planet marked for comparison (grey tick, at 0.71). TRAPPIST-1b: 503 ± 26 K against a bare-rock 508 K; TRAPPIST-1c: 380 ± 31 K against a bare-rock 434 K; LHS 3844b: 1040 ± 40 K against a bare-rock 1031 K; GJ 367b: 1728 ± 90 K against a bare-rock 1741 K. Three sit at the bare-rock limit within their errors, which leaves room for no thick atmosphere on dark rock. TRAPPIST-1c sits about one in eight below it, which a thin atmosphere could explain, and so could a moderately reflective surface. None comes near the even-redistribution value that a thick atmosphere would give.

A day side as hot as bare rock

A planet with no air can be no hotter on its day side than bare rock that re-radiates every photon where it lands. An atmosphere carries some of that heat to the night side, so the day side comes out cooler. For a rocky planet too small for its spectrum to be read, that difference is the test: three of the first four small planets around red dwarfs measured this way sit at the bare-rock limit, and the fourth sits below it by an amount that a thin atmosphere, or a brighter surface, could explain equally well.

exoplanets · Exoplanet atmospheres

Named alongside it

The objects these essays reach for when they reach for this one.

Gravity gradientLibrationRoche limitTidal forceAlbedoAngular momentumBlack holeClimate modelHabitable zoneHeat redistributionLunar laser rangingM dwarf

All concepts